3. The attempt at a solution
I think expansion along the energy space is unnecessary, since the original kets are themselves eigenkets of the Hamiltonian. As for position space, is the Fourier transform [tex]\Psi(x) = \frac{1}{\sqrt{h}} \int_{\infty}^{\infty} | \psi(t) \rangle \exp{\left( \frac{i \omega x}{\hbar}\right)} \; d\omega = \langle \phi | \psi(t) \rangle[/tex], where [tex]| \phi \rangle = \exp{\left( \frac{-i \omega x}{\hbar}\right)}[/tex]?

EDIT: Since only projection onto the position-space and momentum-space bases is necessary, would it be prudent to multiply the vector
[tex]
\left[ \begin{array}{c} 3\\
2\\
0\\
0\\
0\\
1
\end{array}\right]
[/tex]
by the matrix representations of the position and momentum operators respectively?